Monte Carlo Methods and Simulating Quarks Michael Creutz Brookhaven - - PowerPoint PPT Presentation

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Monte Carlo Methods and Simulating Quarks Michael Creutz Brookhaven Lab 1946: Stanislaus Ulam use random trials to estimate probabilities 1947: with von Neumann and others Monte Carlo methods for neutron diffusion 1953: Metropolis,

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SLIDE 1

Monte Carlo Methods and Simulating Quarks

Michael Creutz Brookhaven Lab 1946: Stanislaus Ulam

  • use random trials to estimate probabilities

1947: with von Neumann and others

  • Monte Carlo methods for neutron diffusion

1953: Metropolis, Rosenbluth, Teller, Teller

  • ‘‘Equation of State Calculations by Fast Computing Machines’’

1980’s: extensive application to quantum field theories Now the primary source of non-perturbative information for QCD

Michael Creutz BNL 1

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SLIDE 2

Monte Carlo for statistical mechanics

Partition function Z =

i e−βEi

  • a very big sum
  • Ising on a 10 by 10 lattice gives 2100 = 1.3 × 1030 terms
  • age of universe ∼ 1027 nanoseconds

But we rarely need them all Generate a few ‘‘typical configurations’’

  • random with Boltzman weight e−βE(C)

Michael Creutz BNL 2

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SLIDE 3

Algorithms

Detailed balance (sufficient, but not necessary)

  • P(C → C′)e−βE(C) = P(C′ → C)e−βE(C′)
  • guarantees approach to equilibrium
  • if ergodic, eventually will get there

Metropolis algorithm

  • try some random change C → C′
  • accept change with probability min(1, eβE(C)−βE(C′))
  • gives detailed balance
  • adjust size of changes for reasonable acceptance

Michael Creutz BNL 3

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SLIDE 4

Quantum field theory

Fields φ, interactions from an action S(φ)

  • path integral
  • (dφ)eiS(φ)
  • go to Euclidian space
  • evolution with e−Ht instead of eiHt
  • settle to ground state

Path integral mathematically a statistical mechanics partition function

  • Z =
  • (dφ)e−S(φ)
  • coupling g2 ↔ temperature T
  • use the same Monte Carlo method as for stat. mech.

Euclidian space-time

  • 3D quantum field theory equivalent to 4d stat mech

Michael Creutz BNL 4

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SLIDE 5

Control divergences with a lattice

Quark paths or ‘‘world lines’’ − → discrete hops

  • four dimensions of space and time

a

t x

A mathematical trick

  • lattice spacing a → 0 for physics
  • a = minimum length (cutoff) = π/Λ
  • allows Monte Carlo computations

Michael Creutz BNL 5

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SLIDE 6

What drove us to lattice Monte Carlo?

Late 1960’s

  • quantum electrodynamics: immensely successful, but ‘‘done’’
  • eightfold way: ‘‘quarks’’ explain particle families
  • electroweak theory emerging
  • electron-proton scattering: ‘‘partons’’

Meson-nucleon theory failing

  • g2

4π ∼ 15

vs.

e2 4π ∼ 1 137

  • no small parameter for expansion

Michael Creutz BNL 6

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SLIDE 7

Frustration with quantum field theory ‘‘S-matrix theory’’

  • particles are bound states of themselves
  • p + π ↔ ∆
  • ∆ + π ↔ p
  • held together by exchanging themselves
  • roots of duality between particles and forces −

→ string theory What is elementary?

Michael Creutz BNL 7

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SLIDE 8

Early 1970’s

  • ‘‘partons’’ ←

→ ‘‘quarks’’

  • renormalizability of non-Abelian gauge theories
  • 1999 Nobel Prize, G. ’t Hooft and M. Veltman
  • asymptotic freedom
  • 2004 Nobel prize: D. Gross, D. Politzer, F. Wilczek
  • Quark Confining Dynamics (QCD) evolving

Confinement?

  • interacting hadrons vs. quarks and gluons
  • What is elementary?

Michael Creutz BNL 8

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SLIDE 9

Mid 1970’s: a particle theory revolution

  • J/ψ discovered, quarks inescapable
  • field theory reborn
  • ‘‘standard model’’ evolves

Extended objects in field theory

  • ‘‘classical lumps’’ a new way to get particles
  • ‘‘bosonization’’

very different formulations can be equivalent

  • growing connections with statistical mechanics
  • What is elementary?

Field Theory >> Feynman Diagrams

Michael Creutz BNL 9

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SLIDE 10

Field theory has infinities

  • bare charge, mass divergent
  • must ‘‘regulate’’ for calculation
  • Pauli Villars, dimensional regularization: perturbative
  • based on Feynman diagrams
  • an expansion in a small parameter, the electric charge

But the expansion misses important ‘‘non-perturbative’’ effects

  • confinement
  • light pions from chiral symmetry breaking

need a ‘‘non-perturbative’’ regulator

Michael Creutz BNL 10

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SLIDE 11

Wilson’s strong coupling lattice theory (1973)

Strong coupling limit does confine quarks

  • only quark bound states (hadrons) can move

space-time lattice = non-perturbative cutoff Lattice gauge theory

  • A mathematical trick
  • Minimum wavelength = lattice spacing a
  • Uncertainty principle: a maximum momentum = π/a
  • Allows computations
  • Defines a field theory

Michael Creutz BNL 11

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SLIDE 12

Wilson’s strong coupling lattice theory (1973)

Strong coupling limit does confine quarks

  • only quark bound states (hadrons) can move

space-time lattice = non-perturbative cutoff Lattice gauge theory

  • A mathematical trick
  • Minimum wavelength = lattice spacing a
  • Uncertainty principle: a maximum momentum = π/a
  • Allows computations
  • Defines a field theory

Be discrete, do it on the lattice

Michael Creutz BNL 11

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SLIDE 13

Wilson’s strong coupling lattice theory (1973)

Strong coupling limit does confine quarks

  • only quark bound states (hadrons) can move

space-time lattice = non-perturbative cutoff Lattice gauge theory

  • A mathematical trick
  • Minimum wavelength = lattice spacing a
  • Uncertainty principle: a maximum momentum = π/a
  • Allows computations
  • Defines a field theory

Be discrete, do it on the lattice Be indiscreet, do it continuously

Michael Creutz BNL 11

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SLIDE 14

Wilson’s formulation local symmetry + theory of phases Variables:

  • Gauge fields are generalized ‘‘phases’’ Ui,j ∼ exp(i

xj

xi Aµdxµ)

i j Uij = 3 by 3 unitary (U †U = 1) matrices, i.e. SU(3)

  • On links connecting nearest neighbors
  • 3 quarks in a proton

Michael Creutz BNL 12

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SLIDE 15

Dynamics:

  • Sum over elementary squares, ‘‘plaquettes’’

2 1 3 4

Up = U1,2U2,3U3,4U4,1

  • like a ‘‘curl’’
  • ∇ ×

A = B

  • flux through corresponding plaquette.

S =

  • d4x (E2 + B2) −

→

  • p
  • 1 − 1

3ReTrUp

  • Michael Creutz

BNL 13

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SLIDE 16

Quantum mechanics:

  • via Feynman’s path integrals
  • sum over paths −

→ sum over phases

  • Z =
  • (dU)e−βS
  • invariant group measure

Parameter β related to the ‘‘bare’’ charge

  • β =

6 g2

  • divergences say we must ‘‘renormalize’’ β as a → 0
  • adjust β to hold some physical quantity constant

Michael Creutz BNL 14

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SLIDE 17

Parameters

Asymptotic freedom g2

0 ∼

1 log(1/aΛ) → 0 Λ sets the overall scale via ‘‘dimensional transmutation’’

  • Sidney Coleman and Erick Weinberg
  • Λ depends on units: not a real parameter

Only the quark masses! mq = 0: parameter free theory

  • mπ = 0
  • mρ/mp determined
  • close to reality

Michael Creutz BNL 15

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SLIDE 18

Example: strong coupling determined

0.1 0.12 0.14

Average Hadronic Jets Polarized DIS Deep Inelastic Scattering (DIS) τ decays Z width Fragmentation Spectroscopy (Lattice) ep event shapes Photo-production Υ decay e+e- rates

αs(MZ) (PDG, 2008) (charmonium spectrum for input, fermion dynamics treated approximately) Michael Creutz BNL 16

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SLIDE 19

Monte Carlo

Random field changes biased by Boltzmann weight.

  • converge towards ‘‘thermal equilibrium.’’
  • P(C) ∼ e−βS

In principle can measure anything Fluctuations → theorists have error bars! Also have systematic errors

  • finite volume
  • finite lattice spacing
  • quark mass extrapolations

Michael Creutz BNL 17

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SLIDE 20

Interquark force

  • constant at large distance
  • confinement
  • C. Michael, hep-lat/9509090

Michael Creutz BNL 18

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SLIDE 21

Extracting particle masses

  • let φ(t) be some operator that can create a particle at time t
  • As t → ∞
  • φ(t)φ(0) −

→ e−mt

  • m = mass of lightest hadron created by φ
  • Bare quark mass is a parameter

Chiral symmetry: m2

π ∼ mq

Adjust mq to get mπ/mρ (ms for the kaon) all other mass ratios determined

Michael Creutz BNL 19

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SLIDE 22

Budapest-Marseille-Wuppertal collaboration

  • Science 322:1224-1227,2008
  • improved Wilson fermions

Michael Creutz BNL 20

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SLIDE 23

Glueballs

  • closed loops of gluon flux
  • no quarks

++ −+ +− −−

PC

2 4 6 8 10 12

r0mG

2

++ ++

3

++ −+

2

−+ *−+

1

+−

3

+−

2

+− +−

1

−−

2

−−

3

−−

2

*−+ *++

1 2 3 4 mG (GeV)

Morningstar and Peardon, Phys. Rev. D 60, 034509 (1999)

  • used an anisotropic lattice, ignored virtual quark-antiquark pairs

Michael Creutz BNL 21

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SLIDE 24

Quark Gluon Plasma

p p

π π

Finite temporal box of length t

  • Z ∼ Tr e−Ht
  • 1/t ↔ temperature
  • confinement lost at high temperature
  • chiral symmetry restored
  • Tc ∼ 170 − 190 MeV
  • not a true transition, but a rapid ‘‘crossover’’

Michael Creutz BNL 22

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SLIDE 25

Big jump in entropy versus temperature

5 10 15 20 100 200 300 400 500 600 700 0.4 0.6 0.8 1 1.2 1.4 1.6

T [MeV] s/T3 Tr0

sSB/T3

p4: Nτ=4 6 asqtad: Nτ=6

  • M. Cheng et al., Phys.Rev.D77:014511,2008
  • use a non-rigorous approximation to QCD

Michael Creutz BNL 23

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SLIDE 26

The Lattice SciDAC Project

Most US lattice theorists; 9 member executive committee:

  • R. Brower, (Boston U.) N. Christ (Columbia U.), M. Creutz (BNL), P. Mackenzie (Fermilab), J. Negele (MIT), C. Rebbi

(Boston U.), D. Richards (JLAB), S. Sharpe (U. Washington), R. Sugar (UCSB)

Two prong approach

  • QCDOC at BNL
  • commodity clusters at Fermi Lab and Jefferson Lab
  • ∼ 3 × 10 Teraflops distributed computing facility

QCDOC

  • next generation after QCDSP
  • designed by Columbia University with IBM
  • on design path to IBM Blue Gene
  • Power PC nodes connected in a 6 dimensional torus
  • processor/memory/communication on a single chip

Michael Creutz BNL 24

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SLIDE 27

QCDOC places entire node on a single custom chip

Michael Creutz BNL 25

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SLIDE 28

Two node daughterboard 64 node motherboard 128 node prototype DOE/RIKEN 24,576 nodes!

Michael Creutz BNL 26

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SLIDE 29

Fermilab: 600 nodes with 2.0 GHz Dual CPU Dual Core Opterons JLAB: 396 nodes of AMD Opteron (quad-core) CPUs

Michael Creutz BNL 27

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SLIDE 30

Unsolved Problems

Chiral gauge theories

  • parity conserving theories (QCD) in good shape
  • chiral theories (neutrinos) remain enigmatic
  • non-perturbative definition of the weak interactions?

Sign problems

  • finite baryon density: nuclear matter
  • color superconductivity at high density
  • θ = 0
  • spontaneous CP violation at θ = π

Fermion algorithms (quarks)

  • remain very awkward
  • why treat fermions and bosons so differently?

Lots of room for new ideas!

Michael Creutz BNL 28